IMP - An Algebraic Formulation of Static Isotropy and Design of Statically Isotropic 6-6 Stewart Platform Manipulators

7/28/2019 IMP - An Algebraic Formulation of Static Isotropy and Design of Statically Isotropic 6-6 Stewart Platform Manipulators

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An algebraic formulation of static isotropy and design of

statically isotropic 6-6 Stewart platform manipulators

Sandipan Bandyopadhyay Ashitava Ghosal

Department of Engineering Design Department of Mechanical Engineering

Indian Institute Technology- Madras Indian Institute of Science

Chennai 600036, India Bangalore 560012, India

Abstract

In this paper, we present an algebraic method to study and design spatial parallel

manipulators that demonstrate isotropy in the force and moment distributions. We

use the force and moment transformation matrices separately, and derive conditions

for their isotropy individually as well as in combination. The isotropy conditions are

derived in closed-form in terms of the invariants of the quadratic forms associated with

these matrices. The formulation is applied to a class of Stewart platform manipula-

tor, and a multi-parameter family of isotropic manipulator is identified analytically.

We show that it is impossible to obtain a spatially isotropic configuration within this

family. We also compute the isotropic configurations of an existing manipulator and

demonstrate a procedure for designing the manipulator for isotropy at a given config-

uration.

Corresponding author, email: [email protected].

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1 Introduction

Isotropy is one of the common measures of performance of a manipulator. In the case of six-

degrees-of-freedom (DOF) spatial manipulators, the term isotropy is generally used in the

context of kinematics. However, in practice, the concept of twist-wrench duality is used to

analyse the 66 wrench transformation matrix H, to obtain conditions such that this matrixhas identical singular values (see, e.g., [1]). A consequence of this approach is the concurrence

of kinematic and static isotropy, where the later implies the ability of the manipulator end-

effector to resist forces and moments equally wellin all spatial directions. Among the spatial

parallel manipulators, the Stewart platform manipulator (SPM) has been studied by several

researchers for isotropy [1, 2, 3, 4]. However, to the best of our knowledge, no mechanically

feasible, non-singular isotropic configuration has been obtained for a manipulator of this

class. Further, it may be noted that the 3 6 sub-matrices of H pertaining to the forceand moment parts have different physical dimensions for an SPM, therefore the physical

significance of the singular values of H is not clear.

In this paper, we present a formulation for the study of static isotropy. Our approach is to

analyse the above-mentioned force and moment transformation matrices separately. We form

the conditions for the force and moment isotropy in terms of algebraic equations involving the

elements of the respective transformation matrices. We solve these equations in closed form

to obtain a multi-parameter family of kinematically valid configurations showing combined

force and moment isotropy. We also present examples of isotropic configurations for an

existing manipulator, as well as demonstrate the design for isotropy at a given configuration

within the family mentioned above.

The paper is organised as follows: in section 2, we present the general formulation of static

isotropy of a spatial manipulator, followed by its application to the semi-regular Stewart

platform manipulator(SRSPM) in section 3. In section 4, we identify, in closed form, a family

of SRSPM showing combined spatial isotropy. This is followed by a numerical example, where

we find such configurations for an SRSPM. In section 5, we study various notions of isotropy

and discuss their inter-relations. We also prove that it is impossible to attain a spatially

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isotropic manipulator with the family of configurations obtained in section 4. In section 5

we present the method of design for isotropy at a given configuration, and illustrate it with

an example. Finally, in section 6, we present the conclusions.

2 Formulation

In this section, we derive the isotropy conditions of a general manipulator from its wrench

transformation matrix. First we describe the formulation for obtaining the distributions of

the force and moment resultants on the moving platform. We follow the approach presented

in [5] in the context of the linear and angular velocity distributions of the moving platform.

Using this approach, the said distributions are obtained from the solution of eigenproblems

of two symmetric matrices. The conditions for force and moment isotropy are then derived

in terms of algebraic equations involving the coefficients of the characteristic polynomials

associated with the above eigenproblems. We assume in the following that the resultant

force on the top platform, F, and the corresponding moment (referred to the centre of the

moving platform), M, are available via linear maps of the actuator efforts (e.g., leg forces

in the case of a platform-type parallel manipulator), fi.1 Therefore we can write F and M

in terms of the respective equivalent transformation matrices:

F = HFf

M = HMf (1)

We analyse the properties of the above two linear maps using well-known tools of linear

algebra (see, e.g., [6]). Following the general formulation given in Appendix A, we get two

1Obtaining such a map is trivial for purely parallel manipulators. However, for hybrid manipulators,

there can be significant difficulty in taking the reactions at the passive joints into account while computing

the effect of the actuator effort on the end-effector.

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eigenproblems respectively:

gFf = f (2)

gMf = Mf (3)

where gF = HTF

HF and gM = HTM

HM. These eigenproblems have the following charac-

teristics:

The eigenvalues ,M are real and nonnegative.

At the most three of the eigenvalues are nonzero in each case, as the rank of HFor HM can not exceed three. Therefore, if dim(gF) = n with n > 3, at least (n 3)eigenvalues of gF are zeros and the same applies to gM as well.

The characteristic equation of gF may be written with real coefficients ai as:

0 =

2 + a1 + a2, n = 2

3 + a12 + a2 + a3, n = 3

n3(3 + a12 + a2 + a3), n > 3

(4)

The characteristic equation of gM has exactly the same form as above. However, we use

the notations bi for the coefficients, and M for the eigenvalues in that case. From linear

algebra, isotropy of HF and HM imply, respectively:

i = F2 (5)Mi = M2 i = 1, . . . , n

where () indicates the extreme value of a quantity. Under this condition, the force ellip-soid, (the ellipsoid corresponding to F) reduces to a sphere of radius F. Similarly, themoment ellipsoid reduces to a sphere of radius M. This implies that the nontrivial rootsof equation (4) should be equal, and not all of a1, a2, a3 can be zero.

2 The nontrivial roots

2It may be noted here that the coefficients ai, bi can be computed in closed form as described in Ap-

pendix B.

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of equation (4) can also be obtained explicitly in terms of ai using Sridhar Acharyas and

Cardans formul for the quadratic and cubic cases respectively (see, e.g., [7]). However, it

is not required to compute the roots explicitly in order to obtain the conditions for isotropy

from their equality. Instead, those conditions can be easily formed as algebraic equations in

the coefficients ai etc. as follows. For the case n = 2, we equate the discriminant to zero

and obtain the following condition:

a21 4a2 = 0 (6)

For the case n 3, we consider the nontrivial cubic part of equation (4):

3 + a12 + a2 + a3 = 0 (7)

Using the standard transformation = z a13

, the quadratic term may be eliminated to

obtain the cubic in the form: z3 + P z+ Q = 0 (see, e.g., [7]). It is obvious that ifP = Q = 0,

then z = 0, and hence equation (7) has the repeated roots i = a13 , i = 1, 2, 3. In terms ofthe coefficients of the original cubic equation (7), the conditions for equal roots are:

3a2 a21 = 0 (8)2a31 9a1a2 + 27a3 = 0 (9)

Further, if we solve the above equations exactly in the symbolic form, then the second of

them can be simplified using the first, yielding the pair of equations below:

3a2 a21 = 0 (10)27a3

a31 = 0 (11)

The conditions for moment isotropy can be obtained in the same fashion. In the following,

we list down the conditions for the different types of isotropy considered in this paper.

A. Force (F-isotropy): HF is isotropic.

a21 4a2 = 0, n = 23a2 a21 = 027a3

a31 = 0

n 3

(12)

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B. Moment (M-isotropy): HM is isotropic.

b21 4b2 = 0, n = 23b2 b21 = 027b3 b31 = 0

n 3

(13)

C. Combined: Both HF, HM are isotropic. The conditions that apply in this case are

simply the union of the conditions in cases A and B.

a21 4a2 = 0b21 4b2 = 0 n = 2 (14)3a2 a21 = 027a3 a31 = 03b2 b21 = 027b3 b31 = 0

n 3 (15)

3 Isotropy conditions of an SRSPM

In this section, we apply the theory developed in section 2 to formulate the isotropy conditions

of an SRSPM. In addition to its wide-spread technical applications, the other motivations to

choose this manipulator as our example are: (a) it is probably the most well-studied spatial

parallel manipulator (see section 1 for some of the references) (b) no feasible configuration

of any Stewart platform manipulator demonstrating combined static isotropy is reported in

literature to the best of our knowledge. The manipulator along with the frames of reference

used is shown in figure 1. The bottom platform, shown in figure 2, has the legs arrangedin a circle, with each pair lying symmetrically on either side of three axes of symmetry in

the plane. The axes are 23

apart from each other, while the adjacent pair of legs have an

angular spacing 2b. Without any loss of generality, we scale the radius of the circumcircle

of the bottom platform, rb, to unity, thus eliminating one parameter from all subsequent

analysis, and rendering all other length parameters used in this paper dimensionless 3. The

3We use radians for the angular unit in this paper, while the unit for the base radius can be chosen as

convenient.

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6

l

l l

l1

2l3

5 6

b

bb

b

b

1

23

4

5 b

x

y

X

Y

p

O

1

23

4

5 6

l4

a

a a

a

aa

(a) The manipulator

3

4

2

3

b2

Y

X

rb

O

b

b

bb

b

b

1

2

3

4

56

(b) Bottom platform

Figure 1: Geometry of the semi-regular Stewart platform manipulator

top platform geometry is similar, except that the radius of its circumcircle is rt, and the leg

spacing is 2t.

The kinematic constraints defining the manipulator are written in the task-space vari-

ables. The centre of the top platform is described in the base frame asp = (x,y,z)T. The top

platform orientation is described by the matrix R SO(3), where R= Rz()Rx(x)Ry(y)4.The loop-closure equations are written as:

p + Rai bi lisi = 0, i = 1, ..., 6 (16)

where li denotes the length of the ith leg and ai, bi locate the leg connection points with

respect to the platform centres in the respective frames (see figure 1), and si denotes the ith

screw axis along the respective leg. The screw axis can be written in terms of the task-space

variables and actuated variables as:

si =1

li(p + Rai bi), i = 1, ..., 6 (17)

4In this paper, we denote the rotation about the axis X through an angle as Rx() etc.

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The actuation force along the ith leg can be written as Fi = fisi, where fi denotes the sense

and magnitude of the force. In terms of the force transformation matrix, the resultant force

on the top platform can be written as:

F = HFf (18)

where, f = (f1, f2, f3, f4, f5, f6)T is the vector of leg forces, and the matrix HF is given by:

HF =

1l1

(p + Ra1

b1)1 . . .

1l6

(p + Ra6

b6)1

1l1

(p + Ra1 b1)2 . . . 1l6 (p + Ra6 b6)21l1

(p + Ra1 b1)3 . . . 1l6 (p + Ra6 b6)3

where ()i denotes the ith component of the vector . Moment imparted on the top platformdue to the force along the ith leg can be written as Mi = (Rai)fisi. Using the expressionfor si from equation (17), Mi may be written as Mi =

fili

(Rai) (p bi). Therefore theresultant moment M can be written in terms of the moment transformation matrix HM as

M = HMf, (19)

HM =

1l1

(Ra1 (p b1))1 . . . 1l6 (Ra6 (p b6))11l1

(Ra1 (p b1))2 . . . 1l6 (Ra6 (p b6))21l1

(Ra1 (p b1))3 . . . 1l6 (Ra6 (p b6))3

It may be noted that the use of equation (17) ensures that the expressions of HF and HM

are kinematically consistent, i.e., the loop closure equations are automatically satisfied when

they are cast in this form.

The conditions for static isotropy are obtained from HF, HM following the previous

section. The computational steps involved for all the cases A, B, and C are summarised

below.

1. Form the symmetric matrix gF = HTFHF

2. Form the symmetric matrix gM = HTM

HM

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3. Compute the coefficients of the characteristic equations ofgF, gM using equation (45) (see

Appendix B).

4. Use equations (15) or any subset of the same, as appropriate for the different cases of

static isotropy.

4 Analytical results on the isotropy of an SRSPM

We now describe some analytical results for the different cases of isotropy of the SRSPMusing the formulation developed in the last section. The independent variables involved in

the isotropy equations are the position of the top platform p = (x,y,z)T, the orientation

variables , , , and the architecture variables rt, b and t.

4.1 Architecture, configuration constraints

The natural restrictions on the architecture parameters for a mechanically feasible design

would be the following:

rt rt rt where rt, rt > 0 are two prescribed limits. We adopt in this workrt = 1, rt = 1/4.

/3 b, t 0. At both the ends, the hexagonal platforms reduce to triangles, andbeyond these limits the leg connection points with the platforms cross over, and the

legs can interfere mutually.

The moving platform is above the fixed one, i.e., z > 0.

b = t. If the platforms are scaled versions of each other, the manipulator is architec-turally singular [8, 9].

Any solution for the architecture within these restrictions would be termed as feasible or

valid. Other mechanical constraints, such as joint limits, leg limits, and physical dimensions

of the legs etc. are not considered in the present work. As a result, we do not impose

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any ranges on the values of the position and orientation variables, except that z > 0. We

start with the following assumptions which enable us to perform symbolic computations and

obtain exact analytical expressions:

Isotropic configurations and corresponding architectures are obtained only for thecase when the manipulator is in its home position. The home position is defined

as x = y = 0, = = 0. In other words, displacement along and rotation about only

the Z axis is considered.

The leg lengths have special relationships among themselves. We consider a familyof configurations in which alternate legs of the manipulator have equal lengths, i.e.,

length of the odd numbered legs is L1, and that of the even numbered legs L2 = L1,

where > 0 and in general = 1. This choice is motivated by the 3-way symmetryinherent in the manipulator architecture, and the set of configurations is more general

than those studied in [10, 11, 3, 1].

These restrictions by no means reflect any limitation of our formulation; relaxing these has

only the effect of increasing the complexity of problem5.

4.2 Isotropic configurations

To ensure the practical utility of the isotropy, we first check for the possible singularities

within the target family of configurations. The singularities in statics occur when:

DH = detHF

HM

= 0In this case, the condition becomes:

DH =54r3t z

3 cos( )sin()L61

3, = b t

5Although we do not have a proof, we have not been able to find any other family of isotropic configuration

(namely with all unequal leg lengths or at x,y, , = 0) for the SRSPMs studied by us. This is in spite of

extensive numerical searches using various methods.

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From equation (16), we obtain only two distinct equations defining the leg lengths:

L21 = 1 + r2t + z

2 2rt cos 2L21 = 1 + r

2t + z

2 2rt cos(2 ) (20)

Eliminating L1 between the above equations, we get a linear equation in 2, which gives the

positive solution for as:

= (rt cos(2 ))2 + z2 + sin2(2 )

(rt cos )2 + sin2 + z2

The corresponding solution for L1 is obtained as:

L1 =

(rt cos(2 ))2 + z2 + sin2(2 ) (21)

The expressions for , L1 indicate that there are five free parameters, namely rt, , t, and

z, for which the kinematic constraints are valid. We now search for isotropic configurations

within this 5-parameter family of kinematically valid configurations. First, we establish theconditions for isotropy in general.

4.2.1 F-isotropy.

The kinematically consistent F-isotropy conditions computed from equation (12) are found

to share a common factor, which can be written as a polynomial in zF:

c0z4F

+ c1z2F

+ c2 = 0, where (22)

c0 = 2

c1 = 2r2t 4 cos()cos( )rt + cos(4 2) + cos(2)

c2 = (cos(4 2) + cos(2) 2)r2t + 4(cos()sin2(2 )+ cos(2 )sin2())rt + cos(4 2) + cos(2) 2

where zF denotes the height of the centre of the top platform when the manipulator achieves

F-isotropy.

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4.2.2 M-isotropy.

In this case also, the isotropy conditions in equation (13) have a common factor, which is a

quadratic in z2M

:

d0z4M

+ d1z2M

+ d2 = 0, where (23)

d0 = 2

d1 = r2t 2cos() cos( )rt + 1

d2 = r4t + 4 cos() cos( )r

3t 2(cos(2) + cos(2( )) + 1)r

2t + 4 cos()cos( )rt 1

where zM denotes the height of the centre of the top platform when the manipulator achieves

M-isotropy.

4.2.3 Combined static isotropy.

The condition for combined static isotropy is simply the intersection of the above two con-

ditions, i.e., zF = zM. In other words, equations (22,23) should have common root(s) in z2

.The condition for the same can be obtained in closed form by eliminating z2 from these

equations:

c22d20 + c1c2d0d1 + c

21d0d2 c1d1d2 + d22 + c2(d21 2d0d2) = 0 (24)

The eliminant is of degree 6 in rt, but it is possible to write it as rt sin2()sin2( )P5(rt),

where the quintic P5(rt) = 0, as the vanishing of the other factor leads to singularity. The

coefficients of the quintic are functions of the parameters , , and can be derived in closed

form. However, due to their large size, we do not include them here. When the quintic has a

real solution, equations (22,23) share a common root, and the corresponding positive value

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of z can be obtained as:

z =

NzDz

, (25)

Nz = r4t + 4 cos() cos( )r3t 2(2 cos(2( )) cos2() + cos(2))r2t + 2cos()((2 cos(2) 1) cos( )+ cos(3( )))rt + 2 sin2() cos(4 2)

Dz = r2t 2cos()cos( )rt + cos(4 2) + cos(2) 1

4.3 Examples of combined static isotropy

We choose the free parameters as:

b =

5, t =

10, =

18

The numerical values of rt are obtained as:

rt = (0.3789, 0.9828 0.1866i, 1.4795, 5.5939)Of these, only rt = 0.3789 is feasiblefor the ranges of parameters we have chosen. The result-

ing value ofz from equation (25) is obtained as z = 0.4627. The corresponding configuration

is shown in figure 3.

Figure 2: Combined static isotropic configurations of the SRSPM

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5 Impossibility of spatial isotropy within the family of

configurations studied

In this section, we explore the possibility of identifying a configuration showing spatial

isotropy within the three-way symmetric family of configurations studied in this paper. First,

we summarise the different kinds of isotropy discussed in literature and their relationships.

The matrices J and Jv map the active joint rates to the angular velocity and linear

Table 1: Summary of different kinds of isotropy of a six-DOF spatial manipulator

Type of isotropy Condition(s)

1 -isotropy JJT

= 2I32 v-isotropy JvJ

Tv

= v2I33 F-isotropy HFH

TF

= F2I34 M-isotropy HMH

TM

= M2I35 Combined kinematic isotropy Conditions 1 and 2

6 Combined static isotropy Conditions 3 and 4

7a

Spatial isotropy

Condition 5 and JJTv

= 0, or equivalently,

7b Condition 6 and HMHTF

= 0, or equivalently,

7c Conditions 5 and 6

velocity Jv respectively, and denotes the constant norm of the corresponding isotropicentity. As seen in table 1, conditions 1 through 4 are independent, i.e., they can be satisfied

individually without regard to any of the other conditions. Condition 5 was introduced in [5],

while condition 6 is introduced in this paper. As discussed in section 1, condition 7a has been

very popular among researches. However, it has been discussed in detail in [5] that condi-

tions 7 have not been achieved in any physically realisable Stewart platform. Further, it was

also shown conclusively that it was impossible to find a configuration satisfying condition 7a

among a family of configurations that satisfied condition 5. In the following, we show that it

is impossible to satisfy condition 7b within the family of configurations described here that

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satisfies condition 6.

5.1 Impossibility of satisfying condition 7b within the family of

configurations considered

We prove the above following steps similar to those used in [5]. For the sake of brevity, let

the length of the odd-, and even-numbered legs be L and L respectively. With these, the

expression for HMHTF

is obtained as follows:

HMHTF

=

k11 k12 0

k12 k11 00 0 2k11

(26)

where k11 =3z(sin(2)2 sin)rt

2L22, k12 = 3zrt( cos

2+rt2cos(2)+rt)2L22

. Condition 7b would

be satisfied iff k11 = 0 = k12. Setting k11 = 0, we find the positive root for as:

=

sin(2 )sin

(27)

Equating this with the expression for obtained from inverse kinematics, we get:

2 sin( ) cos z2 + r2t + 1 2cos( )rt = 0, sin = 0 (28)Further, substituting the value of from equation (27) in the expression of k12 and setting

it to zero, we get:

3zsin rt (cos cos( )rt)L2 sin(2 ) = 0 (29)

We now have two equations, namely (28,29) in the variables rt and z. To eliminate rt, we

first obtain an expression for the positive solution for rt from equation (29):

rt =cos

cos( ) (30)

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Substituting this value in equation (28), we get an equation in z alone:

2

cos

z2 + cos2 sec2( ) + 1 2cos sin( ) = 0, sin sin(2 ) = 0 (31)Solving the last equation for the positive value of z, we get:

z =

1 cos2 sec2( ) (32)

With these values of , rt, z we calculate the matrix HFHTF

as:

HFHTF

=

3cos2 sinsinL2 cos3()

0 0

0 3cos2 sin sin

L2 cos3()0

0 0 6cos2 sinsin

L2 cos()

(33)

The isotropy condition in this case reduces to the single equation:

cos2( ) = 0 (34)

Therefore, condition 3 is satisfied when we have:

= 4

(35)

With the same values of the parameters, the matrix HMHTM

is calculated as:

HMHTM =

3sin()sin()

L2 cos( ) 1 0 0

0 1 0

0 0 2

(36)It is very clear from the above expression that condition 4 cannot be satisfied for the parame-

ter values for which condition 3 is satisfied. Therefore it is impossible to find spatial isotropy

within the family of configurations under investigation. Further, given the impossibility of

meeting either of conditions 7a, 7b within this family, it is logically impossible to satisfy

condition 7c within the same set. Analytical and numerical attempts (not included here for

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the sake of brevity) to find real, finite solutions to the combined conditions 5, 6 confirm this

conclusion.

We can summarise the findings of this section as follows: it is impossible to obtain a

spatially isotropic SRSPM within the family of configurations considered in this paper 6.

6 Design of an SRSPM for combined static isotropy

The formulation presented in this paper allow us to solve the problems of analysis and

synthesis within the same setup, in addition to studying the isotropic configurations in

general. In this context, by analysis we mean obtaining the isotropic configurations of a

manipulator with a givenarchitecture, and by synthesis, the determination of the architecture

parameters such that the manipulator is isotropic at a given configuration. We present a

few case studies below.

6.1 Synthesis of an SRSPM for combined static isotropy at a given

position z0 and orientation 0

In this case we assume that the top platform location and orientation have been completely

specified by zF = zM = z0 and = 0 (in conjunction with the assumptions defining the

isotropic family). The task is to obtain and rt such that the manipulator exhibits combined

static isotropy.

We start with the F-isotropy equation (22) and the M-isotropy equation (23). Substi-

tuting the actual expressions of ci, di in these equations, and rewriting them as polynomial

equations in rt, we get a quartic and a quadratic respectively:

g0r4t + g1r

3t + g2r

2t + g3rt + g4 = 0

h0r2t + h1rt + h2 = 0 (37)

6In theory, such configurations may exist outside the family studied here. However, extensive numerical

searches have failed to obtain any solution to this problem.

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The common root of these two equations can be obtained in terms of the coefficients gi, hi

when the resultant with respect to rt vanishes. The resultant is a complicated expression

involving trigonometric terms in , and algebraic terms in z0. We transform it to a polynomial

in t = tan(/2) and simplify its coefficients using the algorithms described in [9]. This results

in a 32nd degree polynomial in t. Extracting the real values of t such that the corresponding

values of are within the prescribed limits, we compute rt numerically. For every positive

solution for rt within the specified range, the free parameter t can be chosen as convenient,

and the architecture of the manipulator can be completely prescribed. We illustrate this

synthesis procedure with an example below.

We choose the configuration as z0 = 1/2, 0 = /20, and the free architecture parameter

as t = /12. Corresponding to these numerical values, there are 24 real solutions for t, of

which, however, only 2 turn out to be feasible. The feasible values of are (0.1750, 0.3321)and the corresponding values of rt are (0.3239, 0.3239) respectively. The configurations are

shown in figures 3(a)-3(b).

(a) = 0.1750, rt = 0.3239 (b) = 0.3321, rt = 0.3239Figure 3: Combined static isotropy of an SRSPM at a given location and orientation

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6.2 Combined static isotropic configurations of an SRSPM of given

architecture

In this section, we find out the configurations of an SRSPM of given architecture showing

combined static isotropy. The manipulator geometry is completely specified in terms of the

architecture variables, rt, and t. We need to find the configuration variables z and such

that the conditions for combined static isotropy are met.

We refer to the condition for combined static isotropy in equation (24), which is a function

of alone. We convert this equation into a polynomial in u = tan(/2) using the symbolic

simplification tools as in the case of the synthesis. In this case we end up with a 8th degree

polynomial in u. For each of the feasible values of arising from the solutions for u, the

corresponding value of z can be computed uniquely from equation (25), thereby completing

the specification of the manipulator configuration. We demonstrate the solution procedure

with an example below.

We use an architecture based on the INRIA prototype of the SRSPM (data adopted

from [12]). However, we take the top platform to be the moving one. In our notation, the

architecture parameters are calculated as:

rt = 0.5803, b = 0.2985, t = 0.6573

The isotropic configurations are shown in figures 4(a)-4(b).

7 Conclusion

In this paper, we have developed an algebraic formulation for the study of static isotropy of

spatial manipulators. We have applied the theory to SRSPMs, and obtained in closed-form a

family of configurations showing force and moment isotropy simultaneously. The formulation

allows us to design an SRSPM for combined static isotropy at a given configuration within

this family. It also allows us to obtain such configurations for an existing SRSPM with a

given architecture. The analytical procedures and results presented in the paper have been

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(a) z = 1.0216, = 2.7254 (b) z = 1.0216, = 2.0078

Figure 4: Combined static isotropic configurations of the SRSPM with INRIA geometry

numerically illustrated with examples of both analysis and design. However, it is found that

it is impossible to attain spatial isotropy within the family of SRSPM showing combined

static isotropy.

8 Acknowledgment

The authors wish to thank the anonymous reviewers for their comments, which have helped

improve the paper.

References

[1] K. Y. Tsai and K. D. Huang, The design of isotropic 6-DOF parallel manipulators

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A Real linear maps and geometry of the range space

Let us consider the real linear map f : Rm Rn which takes x to Ax. We wish to obtainthe distribution of all possible y = A x for any x Rm. This can be done by studying thepossible values of norm of y, which can be computed from the following:

y2 = xTATAx = xTQx (38)

where Q = QT Rmm. Without any loss of generality, we study the maps of the unitsphere Sn1 alone7, i.e., we restrict ourselves to x = 1. The extreme values of y2 cantherefore be obtained by obtaining the extrema of the following function:

h(x) = xTQx + (1 xTx) (39)

where represents Lagranges undetermined multiplier. After differentiation with respect

to x and rearrangement, we obtain the following eigenproblem:

Qx = x (40)

It is well known in linear algebra(see, e.g.,[6]), that R and 0 as Q is symmetric, andx Rm. If (i , xi ) form an eigen-pair satisfying equation (40), extreme values ofy occurwhen:

yi = Ax

i i = 1, . . . , n (41)

and the extreme values are:

yi =

xiTQxi =

i i = 1, . . . , n

Finally, one can write:

yi =

i x

i (42)

7The results obtained thus can be appropriately scaled when

x

= 1.

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Equation (42) allows a direct study of the conditioning of the matrix A (and therefore

the nature of the map f), including defining the exact criteria for the two extreme cases:

singularity and isotropy. If all i = 0 then it is well known that y lies on a n-dimensional(hyper)-ellipsoid, whose axes are defined by yi , i.e., axes lengths are given by 2

i and

corresponding axes are aligned with yi . Ifm < n, then (n m) eigenvalues are always zero,and the dimension of the ellipsoid is restricted to m. However, ifp additional eigenvalues are

zeros, the ellipsoid loses p dimensions, and reduces to an ellipse, a line through the origin,

and the origin respectively when (min(m, n)

p) = 2, 1, 0. The matrix is termed singular

when one or more of is vanish.

On the other extreme, if i = j (= say), i = j, then the the ellipsoid cuts the planesparallel to yi, yj in circles of radius . Similarly, if all the eigenvalues are equal , then

the ellipsoid reduces to the sphere Sn1 scaled by

. Under this condition, the matrix Q

equals In, where In denotes the n n identity matrix. The singular values of A equals iin general, and in this case, the condition number of A, defined as the ratio of the largest

singular value to the smallest one, is seen to reduce to unity. The matrix is termed isotropicin this case.

B Symbolic construction of the characteristic polyno-

mial of a square matrix

The characteristic polynomial of a square matrix A of dimension n is defined as

Pn() = det(In A)= n + a1

n1 + + an1 + an(43)

However, construction of the polynomial from this definition requires symbolic expansion of

the determinant, which is computationally very expensive, and can indeed be prohibitive.

Fortunately, we can compute the coefficients ai above directly using a simple formula derived

below from the Newtons identity.

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Let sk = ni=1 ki , k = 1, . . . , n, where i is a root of equation (43). Then for k = 1, . . . , n,Newtons identity states that, for k = 1, . . . , n,

sk + a1sk1 + a2sk2 + + an1s1 + kak = 0 (44)

Noting that ifAx = x, then for any positive integer i, we have Aix = ix, and that for any

square matrix the trace equals the sum of its eigenvalues, we get the relation sk = tr (Ak),

k = 1, . . . , n. Finally, substituting sk in equation (44) we get an explicit formula for ak as

follows:

ak =

tr (A) k = 1(1)k

tr (Ak) +

k1i=1 aitr (A

ki)

k = 2, . . . , n

(45)

The above method involves mostly the computation of the traces of Ak, and it is therefore

much more economical than the explicit expansion of the determinant. The complexity of

the algorithm can be further reduced by taking advantage of any special structure of A, such

as bandedness, symmetry or skew-symmetry.

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IMP - An Algebraic Formulation of Static Isotropy and Design of Statically Isotropic 6-6 Stewart Platform Manipulators